Characteristic-potentially characteristic subgroup: Difference between revisions

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
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This term is related to: NPC conjecture
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Definition

Symbol-free definition

A subgroup of a group is termed characteristic-potentially characteristic if there is an embedding of the bigger group in some group such that, in that embedding both the group and the subgroup become characteristic.

Definition with symbols

A subgroup $H$ of a group $G$ is termed characteristic-potentially characteristic in $G$ if there exists a group $K$ containing $G$ such that both $H$ and $G$ are characteristic in $K$ .

Formalisms

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In terms of the upper-hook operator

Given two subgroup properties $p$ and $q$ , the upper-hook operator of $p$ and $q$ is defined as the following property $r$ : a subgroup $H$ of a group $K$ has property $r$ if there exists a group $G$ containing $K$ such that $H$ has property $p$ in $G$ and $K$ has property $q$ in $G$ .

The property of being strongly potentially characteristic is thus obtained by applying the upper-hook operator to the property characteristic subgroup with itself.

Relation with other properties

Stronger properties

Characteristic subgroup

Weaker properties

Normal-potentially characteristic subgroup
Normal-potentially relatively characteristic subgroup
Potentially characteristic subgroup: For proof of the implication, refer Strongly potentially characteristic implies potentially characteristic and for proof of its strictness (i.e. the reverse implication being false) refer Potentially characteristic not implies strongly potentially characteristic.
Normal subgroup: For proof of the implication, refer Characteristic-potentially characteristic implies normal and for proof of its strictness (i.e. the reverse implication being false) refer Normal not implies characteristic-potentially characteristic.
Potentially relatively characteristic subgroup: This is a weaker notion, that turns out to be the same as normality. For full proof, refer: Potentially relatively characteristic equals normal
Characteristic-extensible automorphism-invariant subgroup
Normal-extensible automorphism-invariant subgroup

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

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Intersection-closedness

The problem of whether an intersection (finite or arbitrary) of subgroups with this property again has this property is an open problem.

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@@ Line 7: / Line 7: @@
 ===Symbol-free definition===
-A subgroup of a group is termed '''strongly potentially characteristic''' if there is an embedding of the bigger group in some group such that, in that embedding both the group and the subgroup become characteristic.
+A subgroup of a group is termed '''characteristic-potentially characteristic''' if there is an embedding of the bigger group in some group such that, in that embedding both the group and the subgroup become characteristic.
 ===Definition with symbols===
-A subgroup <math>H</math> of a group <math>G</math> is termed ''potentially characteristic'' in <math>G</math> if there exists a group <math>K</math> containing <math>G</math> such that both <math>H</math> and <math>G</math> are characteristic in <math>K</math>.
+A subgroup <math>H</math> of a group <math>G</math> is termed '''characteristic-potentially characteristic''' in <math>G</math> if there exists a group <math>K</math> containing <math>G</math> such that both <math>H</math> and <math>G</math> are characteristic in <math>K</math>.
 ==Formalisms==
@@ Line 31: / Line 31: @@
 ===Weaker properties===
-* [[Stronger than::Semi-strongly potentially characteristic subgroup]]
+* [[Stronger than::Normal-potentially characteristic subgroup]]
-* [[Stronger than::Semi-strongly potentially relatively characteristic subgroup]]
+* [[Stronger than::Normal-potentially relatively characteristic subgroup]]
 * [[Stronger than::Potentially characteristic subgroup]]: {{proofofstrictimplicationat|[[Strongly potentially characteristic implies potentially characteristic]]|[[Potentially characteristic not implies strongly potentially characteristic]]}}
-* [[Stronger than::Normal subgroup]]: {{proofofstrictimplicationat|[[Strongly potentially characteristic implies normal]]|[[Normal not implies strongly potentially characteristic]]}}
+* [[Stronger than::Normal subgroup]]: {{proofofstrictimplicationat|[[Characteristic-potentially characteristic implies normal]]|[[Normal not implies characteristic-potentially characteristic]]}}
 * '''Potentially relatively characteristic subgroup''': This is a weaker notion, that turns out to be the same as normality. {{proofat|[[Potentially relatively characteristic equals normal]]}}
 * [[Stronger than::Characteristic-extensible automorphism-invariant subgroup]]
@@ Line 48: / Line 48: @@
 {{fillin}}
-==Property operators==
-===Left transiter===
-Every characteristic subgroup of a strongly potentially characteristic subgroup is strongly potentially characteristic. In fact, the same supergroup works.
-That is, suppose <math>M \le G \le H</math> with <math>M</math> characteristic in <math>G</math> and <math>G</math> strongly potentially characteristic in <math>H</math>. Then, there exists a group <math>K</math> containing <math>H</math> such that both <math>G</math> and <math>H</math> are [[characteristic subgroup|characteristic]] in <math>K</math>. Then, we also have that both <math>M</math> and <math>H</math> are characteristic in <math>K</math>, and hence <math>M</math> is strongly potentially characteristic in <math>H</math>.